IZA DP No. 854

Income Satisfaction Inequality and its Causes

Ada Ferrer-i-CarbonellBernard M. S. Van Praag

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Forschungsinstitutzur Zukunft der ArbeitInstitute for the Studyof Labor

August 2003

Income Satisfaction Inequality

and its Causes

Ada Ferrer-i-Carbonell AIAS, University of Amsterdam

Bernard M. S. Van Praag University of Amsterdam, Tinbergen Institute

and IZA Bonn

Discussion Paper No. 854 August 2003

IZA

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IZA Discussion Paper No. 854 August 2003

ABSTRACT

Income Satisfaction Inequality and its Causes∗

In this paper, the concept of Income Satisfaction Inequality is operationalized on the basis of individual responses to an Income Satisfaction question posed in the German Socio-Economic Panel (GSOEP). Income satisfaction is the subjective analogue of the objective income concept and includes objective income inequality as a special case. The paper introduces a method to decompose Income Satisfaction Inequality according to the contributions from variables such as income, education, and the number of children. Given the panel structure of the data, inequality may be attributed partly to permanent individual circumstances and partly to transitory changes. The paper shows that by far the largest part of the satisfaction inequality has to be ascribed to unobserved heterogeneity. Distinguishing between a structural and an unexplained part of inequality we find that income explains the largest part of structural Income Satisfaction Inequality together with household membership; for non-working individuals, the age distribution is very relevant as well. JEL Classification: D63, I32 Keywords: equivalent income, financial satisfaction, income satisfaction, income

inequality, variance decomposition Corresponding author: Bernard M.S. van Praag Faculty of Economics and Econometrics University of Amsterdam Roetersstraat 11 1018 WB Amsterdam The Netherlands Tel.: +31 20 525 60 18 Fax: +31 20 525 60 13 Email: B.M.S.vanPraag@uva.nl

∗ The authors would like to thank Peter Hop and Adam Booij for their help and comments. The usual disclaimers apply.

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1. Introduction

Since Gini (1912) and Dalton (1920), the distribution and inequality of income has

been an important subject of study for economic and social scientists. Recent surveys

are offered in the handbooks edited by Atkinson and Bourguignon (1999) and Silber

(1999). The study of income inequality entails two main issues (see Cowell (1999)).

First, the income concept has to be operationalized and measured. Second, a definition

of inequality has to be agreed upon and consequently an index of inequality, namely a

measure of the dispersion of income or welfare, has to be chosen.

The basic question underneath is why we are so interested in income inequality.

It is not just an administrative statistic. The reason is that income or ‘equivalent

income’ is taken as a proxy for welfare. It follows that income inequality is seen as

synonymous to welfare inequality, a performance index of society. The literature

bears witness that there is no generally accepted measure of welfare. This is caused

among other reasons by uneasiness about whether income in itself is a suitable

measure of welfare. This is especially true for modern welfare states where a

considerable part of our consumption is provided by the state and not through the

market. Additionally, income has to be corrected for individual and household

characteristics if it aims at measuring welfare. For example, it is evident that two

households with the same income but different family sizes fs will need different

incomes to be equally satisfied. Hence, income y should be ‘corrected’ for family size,

which would lead to what is known as ‘equivalent income’ y� . For instance, if we

apply the correction factor g(fs) equivalent income becomes . ( )y y g fs=� . In order to

compare incomes and to get some idea about income inequality, it does not make

much sense to define income inequality on nominal income. Inequality should be

measured with respect to equivalent incomes. It is evident that this will change

income inequality. For instance, if we use the variance of log-incomes as inequality

measure, we have

var(ln( )) var(ln( )) var(ln( ( )) 2cov(ln( ), ln( ( )))y y g fs y g fs= + +� (1)

This shows that the inequality of equivalent incomes, as defined by the log-variance,

will differ from the inequality of nominal incomes. One may be larger or smaller than

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the other. It is evident that there are more intervening variables than family size. Say,

we have a vector x of such variables. Then the definition of equivalent income may be

straightforwardly generalized to correct for x. The problem is the definition and

empirical operationalization of the correction factor g(x). Our definition will be

simple. If two individuals A and B report to be equally satisfied with their incomes yA

and yB , we assume those two incomes to be equivalent. This implies that if we take A

as the reference then ( ) B Ag x y y= . In practice, this factor is derived by looking at

’income satisfaction questions’, which are now posed as a matter of routine in socio-

economic surveys. It may also be that x includes some unobservable individual

characteristics ε . Then exact correction at the individual level will be impossible.

However, we can still operationalize for those variables that are observable.

In this paper, we try a new approach to assess inequality by not looking at

nominal income as our basic variable but at the satisfaction derived from income. We

call this income satisfaction and we measure it by means of individual answers to an

income satisfaction question. We believe that this empirical approach leads to a

welfare inequality concept, which does more right to our intuitive feelings about

inequality than the measures that account for income differences only.

It is possible to define and measure an index for income satisfaction. The

income satisfaction concept used in this paper does implicitly incorporate the

necessary corrections. Income satisfaction is empirically defined through the analysis

of individual responses to an income satisfaction question. The paper aims at

explaining the individual’ s income satisfaction by objective variables x, such as

income, education, and number of individuals in the household. We denote that

satisfaction by ( ; , )f y x θ , where y stands for nominal income, x for other individual

circumstances, and θ for a vector of parameters to be estimated. If there

holds ( ; , )f y x yθ ≡ , the subjective perception will coincide with nominal income.

Hence, usual income inequality is embedded in the income satisfaction inequality

concept as a special case.

The income satisfaction inequality (Isat) is here measured in such a way that if

satisfaction would coincide with income, that is yxyf ≡),;( θ , then (Isat) would equal

the variance of log-incomes. The variance of the logarithm is one of the most

frequently used measures of inequality, together with the relative mean deviation, the

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variance, the coefficient of variation, the Atkinson index, the Gini coefficient, and

Theil’ s entropy measure (see Atkinson, 1970; Sen, 1973). All those inequality

measures are functions of moments of the income distribution. It is well known that

the distribution of personal and household incomes is rather well –approximated by a

log- normal distribution. When the income distribution is (approximately) log-normal

),( 2σµΛ , all income inequality indexes are (approximately) functions of the two

distribution parameters. The parameter µ gives the position of the distribution, while

2σ is a measure for the relative income differences. The log-variance ( 2σ ) as a

measure of inequality has the advantage that it does not depend on the money unit.

Other measures are simple functions of 2σ and µ . Theil (1967, chapter 4; 1979)

shows that the Theil Entropy measure equals (½) 2σ in the case of log-normality. Van

Praag (1978) derived a similar result for the Atkinson index, and Aitchison and

Brown (1960) for the Gini index. If approximate log-normality holds, there is not

much to be gained by considering more indices simultaneously, given the one-to-one

relationship between such indexes. Thus we will exclusively focus on the variance of

the logarithm. This choice is clearly a subjective one but this index is very useful

when looking at the causes of inequality. It is obvious that we may apply to the

income satisfaction ),;( θxyf any inequality index that may be applied to nominal

income y. This holds for the Atkinson index and the related social welfare function

(SWF-) approaches. Let the SWF be defined as ( ( , ; ))n nn

h f y x θ∑ , where h(.) stands

for the contribution to the SWF of an individual with individual satisfaction f. Then

the SWF-maximizing situation is found by maximizing the SWF under the constraint

nn

y Y=∑ , where Y stands for (fixed) national income. Under general circumstances

this optimum will be reached if individual satisfactions (not incomes !) are equal.

The paper focuses on the study of the causes of income satisfaction inequality

(Isat). This is equivalent to examining which objective variables contribute most to the

existing income satisfaction variance. Since individual income satisfaction can be

partly explained by differences in income, the number of children, age, and education,

income satisfaction inequality can be decomposed along the same lines. Thus, Isat is,

to a certain extent, explained by the underlying inequalities in those objective

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variables. If we have longitudinal data, income satisfaction inequality can be further

decomposed according to individual permanent differences in objective factors and

individual transitory changes. Finally, income satisfaction inequality can also be

decomposed into within-and between-group inequalities. We consider the inequality

between East and West Germany and between the groups of workers and non-

workers.

The novel contribution to the literature of the present approach to measure

income satisfaction inequality is threefold. First, if individual satisfaction with own

income is not only caused by income, but is also dependent on other individual

characteristics, such as age and family size, the income satisfaction concept implicitly

includes the corrections required to make individual welfare equivalent and

comparable. Second, the empirical estimation of income satisfaction allows for testing

different specifications of the relationship between income and income satisfaction.

As income inequality has not to be seen as an administrative index but as an index of

the inequality in income satisfactions, we need the best possible specification of that

relation between income and income satisfaction. Third, as we know the causes of

income satisfaction, we may also view satisfaction inequality as caused by

inequalities with respect to the underlying variables like income, age, household size,

and so on.

This paper is organized as follows. Section 2 introduces the data and the income

satisfaction question. Section 3 presents the estimation results for the income

satisfaction question. Section 4 discusses the income satisfaction inequality concept,

the decomposition method, and presents the empirical findings on the causes of

inequality. Section 5 concludes.

2. Income satisfaction

The empirical analysis is based on the German Socio-Economic Panel (GSOEP) data.

The GSOEP is a longitudinal household panel that started in the Federal Republic of

Germany in 1984. After the reunification of Germany, (former) East-German

households have been included (see Wagner et. al, 1993). This paper is based on the

waves 1992 to 1997, including more than 20,000 individuals of which about 30% are

Eastern individuals. It is well-known that the two parts of the country have lived since

1945 under different regimes with respect to opportunities, fiscal and social security

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regulations, the supply of public goods and housing, and last but not least political

philosophy. Although the populations are converging since the reunification of the

two German states, we think it still prudent to handle them as different sub-samples

for the period 1992-7. Further, the sample is divided between workers and non-

workers, as we assume that the significance of income will differ between workers

and non-workers. For workers the income concept will have two significances. The

first one is that of income as a source of material well-being. The second meaning is

derived from the fact that the income level is strongly correlated with how your work

contribution is evaluated by the employer and the social environment. For individuals,

who live from pensions, social benefits or alimonies the second meaning of income is

not or hardly relevant. Each group will also refer itself to a different reference group.

It is known that many people without a proper job environment feel rather isolated

from what is going on in society at large. The set of non- workers is somewhat

heterogeneous, since it includes unemployed workers, housewives, who have no

intention to participate in the labor force and retired individuals. More precisely, the

Western group consists of 65% women, 23% men younger than 65 and 12% males

older than 65. For the Eastern non-working population the corresponding figures are

62%, 28% and 10%.

From table 2 we will see whether these distinctions are empirically justified.

Since the numbers of individuals who switch from East to West, or from ‘non-

working’ to ‘working’ , and vice versa are very small, they are treated as new

respondents in the new group (see Hunt, 1999, 2000; Pannenberg, 1997).

The Income Satisfaction (IS)- question is asked to all respondents of the

GSOEP. Satisfaction questions have appeared in questionnaires for over more than

three decades starting with Cantril (1965) and Likert (1932). The Income Satisfaction

question in the GSOEP that is used in this paper runs as follows

’How satisfied are you today with the following areas of your life?

(Please answer by using the following scale, in which 0 means totally unhappy and 10 means

totally happy)

How satisfied are you with your household income ……………………………’

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The answer to this question is termed the individual’ s Income Satisfaction (IS) level.

In this module the discrete answers vary from 0 to 10, where 0 stands for ‘totally

unhappy’ and 10 for ‘totally happy’ . Satisfaction questions have been amply used by

economists, psychologists, and sociologists. Economists have used answers to

satisfaction questions as a proxy measure of the individual’ s welfare in order to study

individual preferences, behavior, welfare, and poverty (see, for example, Clark and

Oswald (1994), Frey and Stutzer (2002), Plug, Van Praag, Krause and Wagner (1997),

DiTella et al., 2001; Easterlin, 2000; Ferrer-i-Carbonell and Van Praag, 2001; Frijters,

2000; Frey and Stutzer, 2000; Ng, 1997; Van Praag, 1971; Van Praag et al., 2001).

In order for IS questions to be meaningful, one needs to assume that respondents

are able to understand and to answer subjective questions and that they evaluate and

respond to such questions in a similar manner, such that individual answers can be

compared. This does not necessarily imply that individuals who grade their

satisfaction at 8 are twice as satisfied as those who grade their satisfaction by 4.

Interpersonal comparability does not imply a cardinal interpretation per se. The

literature on subjective well-being, which is large and growing (for an overview see

Kahneman et al. 1999 and Diener et al., 1999) shows clear consistencies across

studies. This may be interpreted as empirical evidence of the meaningfulness of

questions on satisfaction and of the capacity and willingness of individuals to respond

to such questions. The assumption of interpersonal comparability has been long

discussed in the literature (see, for example, Sen, 1999 and Van Praag, 1991). In this

paper, we start from the working hypothesis that individual answers to satisfaction

questions are (ordinally) comparable among individuals. Thus, it is assumed that two

individuals, answering a ‘5’ , experience the same level of income satisfaction,

although their material circumstances may differ. Notwithstanding the fact that there

is a vast amount of literature, which either implicitly or explicitly starts from this

comparability assumption, it is impossible to provide direct evidence that this

assumption is warranted. That would require the existence of a basic and generally

accepted method of direct (e.g. physical) satisfaction measurement, but such a method

does not exist (yet). If we ask other questions we find indeed that one question is

correlated with or predicts the results of another instrument quite well. Obviously, but

this is a technical aspect, in practice comparability is always approximate, as the

discrete scaling implies a rounding-off error for each response.

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Objective variables are not the only determinants of individual satisfaction.

Personal traits, such as extroversion, optimism, or capacity to adapt to adverse

situations, are also important determinants of individual’ s welfare. In fact, it is argued

that only about 8 to 20% of individual life satisfaction, which is an even broader

concept than income satisfaction, is explained by objectively measurable variables

(Argyle, 1999; Diener et al., 1999; Kahneman et al., 1999). It is also important to bear

in mind that the individual is subject to adaptation phenomena and the relative income

hypothesis. Adaptation theory suggests that individuals adapt their satisfaction norms

to new situations (Helson, 1964). This phenomenon is called ‘the hedonic treadmill’

by Brickman and Campbell (1971), while Van Praag (1971) coined it ‘preference

drift’ . The relative income hypothesis says that the individual’ s satisfaction with

income depends on how its income compares to that of others (Kapteyn and Van

Herwaarden, 1980, Clark and Oswald (1996), Ferrer-i-Carbonell, 2002)). Thus,

changes in one’ s income or in the income distribution of a society will not necessarily

be reflected into changes in income satisfaction . This has an ethical dimension that is

not further discussed in this paper.

Table 1 presents the distribution frequencies of IS in the total sample. We see

that the bulk of the population is found in the classes 4 to 8, but there are also a

substantial number of observations in the extremes. It is especially remarkable that

only about 0.5 % of the respondents evaluate their own financial situation by zero. We

see that the average difference in satisfaction between Western workers and non-

workers is not large, but that the difference between the Western and Eastern part of

the country is much larger. The same pattern is found when we look at Table 1b., in

which we tabulated the satisfactions, differentiated according to income quartiles.

Table 1b shows that IS is on average larger, the richer an individual is. This holds

within the sub-samples. For example, the richer 6% of West workers have, in average,

a IS of almost 8, while the poorest 25% of the sample have a satisfaction level of 6.6.

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Table 1.a Frequency distributions and averages of Income Satisfaction (IS), GSOEP 1992-1997 IS West

Workers East Workers West Non-

Workers East Non-Workers

0 0.56% 0.59% 0.78% 1.33% 1 0.49% 0.52% 0.81% 1.24% 2 1.16% 2.16% 1.94% 3.12% 3 2.48% 4.21% 3.65% 6.28% 4 4.05% 6.15% 4.48% 7.74% 5 10.90% 18.02% 12.29% 19.29% 6 11.42% 16.62% 10.39% 13.86% 7 20.75% 23.25% 17.80% 17.07% 8 27.41% 20.28% 24.67% 19.44% 9 12.67% 5.70% 12.04% 6.22% 10 8.11% 2.51% 11.14% 4.41% Average 7.092 6.332 6.992 6.120 Total observations

30539 11360 20611 8501

Table 1.b Averages of Income Satisfaction (IS) per income percentile1, GSOEP 1992-1997 Lowest 25% Lowest 50% Lowest 75% Top 6 % West Workers

Average Income 3011 4105 5432 12088 Average IS 6.633 6.989 7.224 7.976

East Workers Average Income 2697 3463 4188 8955

Average IS 5.769 6.202 6.366 7.693 West Non-Workers

Average Income 2276 3294 4675 10938 Average IS 6.384 6.876 7.214 7.898

East Non-Workers Average Income 1808 2566 3463 7332

Average IS 5.598 6.006 6.387 7.448

3. Estimation

Satisfaction questions are usually explained by means of latent variable models

because IS is an ordered categorical variable. In our case it takes the values 0, 1,…,

10. We assume the usual Ordered Probit model. The real axis is partitioned in

intervals ( ] ( )∞∞− ,,...,, 100 µµ , such that the latent variable ( ]*1,i iIS µ µ +∈ if IS = i.

We assume that the latent variable IS* obeys the equation

1 Due to rounding off the brackets contain approximately 25%.

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*( )nt t y nt z nt x nt y n x n nt nLn IS C y Z X y Xα α α β β ε ν= + + + + + + + (2)

where n stands for the individual and t for time. The explanatory variables are divided

into two groups, i.e. X and Z. The first are included in the regression in two forms: at

the yearly value, Xnt, and as the average of Xn across time t ( nX ). The vector Xnt includes

number of children and adults in the household. The vector of explanatory variables Z are

only included at their yearly value (Znt). As Equation (2) shows, income is included both at its

yearly value and as an average across the 6 years period.

Income satisfaction is assumed to depend on individual and household objective

characteristics. Next, the specification is discussed (see also van Praag et al., 2003).

We assume that satisfaction depends on log-household income y, and the number of

individuals to be supported, where we distinguish between children (below 16) and

adults. As we assume that the number of children will have a negative effect on

income satisfaction, but that this effect will be less negative, the higher the household

income, we introduce an interaction term between income and children as well. Age is

included as log and log-square, because it is frequently found that satisfaction, not

only with income but also with the job, has a U-shape effect on satisfaction. It implies

that as we grow older our needs increase. We become less satisfied with income if it

stays the same over time. This has to do with adaptation and with the fact that most

members of our reference group will enjoy income growth over time. Due to the

squared term the effect reaches a minimum about fifty; after that age the effect

reduces. A similar effect with respect to job satisfaction was found by Clark and

Oswald. We assume that living together will have a positive scale effect, as household

chores can be divided over two and there are substantial economies of scale when

living together. On the other hand a family with two breadwinners has less time

available for household production than a traditional household, which is still

frequently found in Germany 1997. Hence, we assume that a two-breadwinner family

will be less satisfied than a one-breadwinner family with the same household income.

Finally, we include time dummies, representing inflation and general ’rising

expectations’ and a gender dummy, which equals one for a male respondent and zero

for a female respondent.

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As discussed above, some of the explanatory variables are included in two

ways, viz. as their mean value and at their annual values. We do this because we

expect that changes in the X-variables will not immediately affect income satisfaction

to the full extent, but that there will be an adaptation process to new circumstances.

The coefficient of the long-term average nX stands for the effect of a permanent

change, while the coefficient of Xnt - nX stands for the immediate short-term effect of a

change. For statistical reasons this specification was advocated by Mundlak (1978).

He interpreted the nX as picking up the correlation between observed individual

characteristics and the individual unobserved effects. In this way, Mundlak aimed at

ensuring orthogonality between X and ν .

Equation (1) can be rewritten as

*( ) ( ) ( ) ( ) ( )nt t y nt n x nt n y y n x x n z nt nt nLn IS C y y X X y X Zα α α β α β α ε ν= + − + − + + + + + + +

(3)

In equation (2) we distinguish for the X-variables a transitory and a permanent effect.

The permanent effect is (α + β), and the transitory effect is α. For some variables, the

permanent effects have a clear interpretation. For example, the effect of mean income

is the permanent income effect (Friedman, 1957).

As we assume a considerable correlation between annual errors, we model the

error as nt nε ν+ . The individual random effect nν and the error term ntε are assumed

to be normally distributed and to be correlated neither with each other nor with the

explanatory variables X and Z. The total residual variance equals 2 2( ) ( )σ ν σ ε+ . The

individual random effect, which varies over individuals n but which is time-constant

per individual, may be interpreted as standing for those individual psychological traits

that are not observed in the data set (unobserved heterogeneity). We notice that in the

literature of the evolution of wages over time some more complex error structures are

implemented (see Lillard and Willis (1978), Baker (1997), Dickens (2000)). These

interesting ideas seem to be outside the scope of this introductory paper, which

focuses on the new satisfaction inequality concept.

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It is well known that identification in the Probit model is only possible by

addition of a normalizing condition, for which we traditionally take σ2(ε)=1. If we

would impose σ2(ε)=2, we would have to multiply all effects by 2 . It is easy to see

that the t- values under both specifications will be the same. However, we may

impose other identifying conditions like setting one of the non-zero effects equal to

one. Say, under the standard-normalization the effect was α, then under the new

normalization all coefficients would have been multiplied by α-1 while the error

variance would be multiplied by α-2. Hence, one can use standard-software and

change the identifying condition after estimation. This indeterminacy implies that the

value of the income satisfaction inequality index, which we are about to define,

depends on the specific normalization chosen. In order to make the satisfaction index

simultaneously comparable between different samples and with the variance of log-

incomes, we re-normalize by multiplying the Probit-estimates with the factor

1/(αy+ βy). For the variances used in the following tables this implies a multiplication

by (αy+ βy)-2. By applying this normalization we ensure that the structural parts of

income satisfaction estimated with different error variances, may be compared with

each other.

Table 2 presents the estimation results for equation (1) as estimated by an

Ordered Probit model with individual random effects. Table 2 shows that we allowed

for the inclusion of a permanent effect and a transitory effect for three variables, i.e.

income, number of children in the household, and number of adults. The income2

effects are all positive and significant. Hence, normalization by division through the

sum of both effects is a valid operation. The transitory income effect for West-

German workers equals 0.261. For Western non-workers, the effect is of the same

order. For Eastern workers and non-workers the transitory income coefficients are

much larger. The income effect also depends on the number of children via the

interaction term income-children. This interaction term has a slight mitigating effect

on the cost of children for Westerners, but it is non-significant for Easterners.

2 Income is the answer to the following question: “If everything is taken together: How high is the total monthly income of all the household members at present? Please give the net monthly amount, in other words after the deduction of tax and national insurance contributions. Regular payments such as rent subsidy, child benefit, government grants, subsistence allowances, etc., should be included. If not known exactly, please estimate the monthly amount”. (answers are in German Marks).

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The age coefficients are all significant, where Ln(IS*) has a U-shape with

respect to age. Western workers reach a minimum income satisfaction at the age of 44

and Eastern workers at 56. For non-workers, income satisfaction attains its minimum

at around 37. Apart from psychological developments over human life there may be

more mundane reasons: households needs increase with the number of children and

spending on consumer durables and housing is more important in the first half of the

life cycle. Later in life children are leaving the household and investments in housing

and furniture have been completed. Another reason for this fall in satisfaction during

the first period in life may be that individuals get used their material status and take it

as a matter of course. The education effect is positive in the West, non-significant for

Eastern workers, and negative for Eastern non-workers. The effect of more education

is ambiguous. On one hand more education leads to more efficient consumption. This

would point to a positive effect on income satisfaction. We find this effect for West-

German citizens. On the other hand more education widens your horizon and will

increase someone’s expectations. This would yield a negative effect. The education

effect is a mixed product. For East-Germany we find that both factors cancel out for

workers, while the widening horizon-effect has the upper hand for East-German non-

workers. The presence of more adults or children has a negative effect on income

satisfaction for all four sub-samples. If one lives together with a partner in one

household, this increases individual income satisfaction. Male respondents are less

content than females. The estimation results show that this coefficient is non-

significant. The individual random effect, i.e. individual unmeasured psychological

characteristics, explains between 30 and 40% of the total unexplained variance, being

somewhat higher for Westerners than for Easterners.

We notice that the equation differences between sub-samples show that it is

indeed justified to estimate the model for the four sub-samples separately.

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Table 2. Income Satisfaction (IS) Regression. Ordered Probit with individual random effects, GSOEP 1992-1997

West Workers East Workers West Non-Workers East Non-Workers Estim. Est/StErr Estim. Est/StErr Estim. Est/StErr Estim. Est/StErr

Constant 5.427 3.461 3.660 1.954 16.445 13.431 19.997 10.647 Dummy for 1992 0.416 12.422 -0.079 -2.054 0.141 3.664 -0.359 -5.521 Dummy for 1993 0.425 12.168 0.174 4.303 0.384 9.791 -0.064 -1.076 Dummy for 1994 0.346 11.362 -0.288 -7.822 0.491 14.820 0.098 1.892 Dummy for 1995 0.300 8.582 0.115 2.861 0.387 10.060 0.141 2.199 Dummy for 1996 0.412 11.157 0.226 5.456 0.351 8.805 0.170 2.627

Ln(age) -5.354 -5.981 -4.223 -3.915 -11.861 -17.887 -12.863 -12.779 Ln(age) ^ 2 0.707 5.614 0.529 3.442 1.641 18.114 1.767 12.847

Minimum age reached at 43.993 54.310 37.101 38.053 Ln(net family income) 0.261 7.139 0.422 8.710 0.240 6.023 0.427 5.786 Ln(years of education) 0.258 3.334 0.009 0.090 0.260 2.943 -0.325 -2.588 Ln(number of adults) -0.176 -4.735 -0.247 -4.651 -0.041 -0.848 -0.152 -1.600 Ln(number of children+1) -0.727 -1.973 -0.408 -0.839 -0.778 -2.197 -0.817 -1.140 Ln(net family income) *Ln(child.+1)

0.077 1.732 0.024 0.394 0.079 1.819 0.082 0.919

Male -0.040 -1.319 -0.073 -2.212 -0.257 -7.392 -0.134 -2.794 Living together 0.211 5.860 0.216 3.969 0.259 8.393 0.100 1.743 Two Earners -0.032 -0.964 -0.083 -1.835 XXX XXX

Mean (Ln(net family inc.) 0.739 13.716 0.578 8.666 0.760 13.659 0.573 6.116 Mean (Ln(children+1)) -0.263 -4.572 -0.330 -4.206 -0.452 -6.662 -0.575 -4.673 Mean (Ln(adults)) -0.206 -3.855 0.003 0.052 -0.268 -3.991 -0.148 -1.341

( )σ ε 1.338 0.966 1.217 1.250

)(νσ 1.056 0.711 1.018 0.830

% of variance due to v 38.37% 35.12% 41.16% 30.61% Number of Observations 30356 11256 20510 8501 Log Likelihood -56603 -21157 -39217.9 -16957 Number of Individuals 8130 3191 6361 2690

4. Income satisfaction inequality

This section presents the concept of income satisfaction inequality (Isat), which is

derived by generalizing the objective income inequality concept. Let us assume two

individuals A and B with incomes yA and yB and personal circumstances XA and XB,

respectively, where X stands for the vector of all relevant variables except income.

Then the incomes yA and yB are equivalent satisfaction-wise, if

),(),( **BBAA XyISXyIS = (4)

14

Or in words, incomes yA and yB are equivalent if individuals A and B are equally

satisfied financially, given their different background circumstances X. The case *( , ) ( )IS y X ln y≡ , where income satisfaction and objective income coincide, is a

special case of the income satisfaction concept. This would be the case if all

coefficients in Equation (2) would have been zero except the income coefficients.

From Table 2, it is clear that other variables than income influence income

satisfaction, i.e., populations with the same objective income distributions may have a

different distribution of income satisfaction.

In contrast to what is sometimes thought most income inequality measures are

based on a cardinal utility concept. The inequality index is defined as

1 1( ( ),..., ( ))N NI I f y f y= . It follows that the impacts of income increases for one

individual are cardinally defined and the same holds for the comparison of income

changes between individuals. We may change the functions 1 1( ),..., ( )N Nf y f y into

monotonic transforms 1 1( ),..., ( )N Nf y f y� � but then the inequality index will be changed

as well. We may think, for instance, on ( )f y y= and ( ) ln( )f y y=� . We may see

*f IS=� as the specification in this paper.

In this paper, we consider *( ( )) satVar Ln IS I= as our inequality measure.3 It is

well known that the variance of log-incomes is not an ideal inequality measure, and

this holds as well for the variance of income satisfactions, but the problem is that

other measures also suffer from problems. There is no ideal measure. One evident

advantage of the variance is its decomposability, which we shall use in this paper. As

pointed out before, we may just as well take another statistic based on *( )Ln IS like

Theil’ s entropy, or the Atkinson index. Within the scope of this paper this would not

add new information.

Table 3 presents estimates of the income satisfaction inequalities in the four

sub-samples, which we compare with the corresponding objective income inequality.

In the first line we present the objective income inequalities. In the second line we

present the corresponding income satisfaction inequalities, defined as the variances of

the structural parts of the estimated income satisfactions. In the third line we present

3 The variance of Ln(IS) was calculated using individual weights as available in the GSOEP data. The weights represent the inverse probability of selection.

15

the structural variance as a percentage of the ’total’ variance, that is, the structural

variance plus the error variance ( 2 2( ) ( )σ ν σ ε+ ). For workers total variance is twenty

times (!) the structural variance while for non-workers the proportion is ten to one.

This is of course due to the small explanatory power of these models in terms of R2 or

pseudo-R2. We may also interpret this result as saying that by far the largest part of

satisfaction inequality is caused by unobserved heterogeneity and/or random errors.

As far as it is caused by random disturbances we cannot improve on this result. As far

as it is caused by unobserved variables we have to look after additional relevant

observable individual characteristics. Both objective income and structural

satisfaction inequalities seem to underestimate the perceived inequality. We have to

accept this as a fact of life. It does not imply that the structural inequalities have

become devoid of interest.

Table 3: Objective and income satisfaction inequalities West Workers East Workers West-Non

Workers East-Non Workers

Variance of objective Log-incomes 0.218

0.173

0.284

0.218

Variance of Log-income satisfactions (structural part)

0.186 0.141 0.357 0.328

structural part as percentage of total variance

6.03% 8.94% 12.41% 12.70%

Total satisfaction variance 3.08 1.57 2.87 2.58 Number of Observations 30356 11256 20510 8501

Table 3 shows that objective income inequality is larger in the West than in the East,

both for workers and for non-workers. We also see that income inequality is larger

within the group of non-workers than for workers. For workers the structural

satisfaction inequality is smaller than that with respect to objective incomes, while the

opposite holds for non-workers. For instance, for western non-workers the objective

inequality is 0.284 and the subjective analogue is 0.357.

Next, we present an income satisfaction inequality decomposition to identify the

contribution of each observable variable X and Z to income satisfaction inequality.

Since the income satisfaction inequality is here defined in terms of variance, studying

the causes of this inequality is equivalent to decomposing the variance of the income

satisfaction. The prototype of a decomposition is based on the model

16

1ln( ) ln( ) ... ln( )kZ Z Z= + +

where we assume that the components are uncorrelated, i.e., cov( , ) 0i jZ Z = . We get

1

2 2 2...kZ Z Zσ σ σ= + +

Hence, we may assign to each component its relative share in contributing to the

overall variance/inequality. The shares are

2

2iZ

iZ

pσ

σ=

The shares add up to one and each share is non-negative. Unfortunately, this

decomposition is impossible if the components are correlated, i.e., cov( , ) 0i jZ Z ≠ .

This is obviously the case for the IS- equation as age, income, education and children

are correlated. Therefore, we have to apply a second-best solution.

The variance decomposition we apply is performed by the well-known stepwise

regression procedure.4 We notice that we defined the structural part of equation (2) as

the right-hand side without error-terms. We denote it by ( )sIS for short. Its variance is

just the structural satisfaction inequality. If we regress ( )sIS on all variables in (2)

simultaneously we get of course an R2 of 100 %. If we regress ( )sIS on one variable

X1 (and a constant) we get an R21 of less than 100 %. It may be interpreted as the

contribution of X1 to total inequality. Adding a second variable X2 the R22 is increased

from R21 to R2

2. The difference (R22-R2

1) may be interpreted as the additional

inequality contribution caused by differences in the variable X2. It is obvious that this

decomposition depends on the order of introduction of the variables except in the

unlikely case that the two variables X1 and X2 are non-correlated. This variance

decomposition method may be generalized to any arbitrary number of variables. The

usual option in standard-software is that the order of successive introduction of the

4 An alternative decomposition would be by means of principal components. However, here we have the disadvantage that it is often difficult to understand what these principal components stand for.

17

variables is chosen, such that at each step the incremental explanation is maximized.

We call this order the maximizing order. Obviously, we may impose any other order

as well. In Table 4.a we present for each of the four sub-samples the inequality

decomposition according to the (sample-specific) maximizing order. For Western

workers we find that a very large part may be explained by the long-term variables.

The variable Mean (Ln(net family income), that is the average log-net family income

over the period 1992-1997 explains 65.63% of the structural inequality. The next

factor Mean (Ln(adults)) gives an additional contribution of 12.21 % and a similar

contribution is given by the number of children (defined as living at home below 16).

The remaining transitional components give only a minor contribution adding up to

about 12 %. For Western non-workers the picture is somewhat similar, although the

long-term contribution is smaller. One transitional factor Ln(age) plays a significant

role as well. For the Eastern part of the population the situation is completely

different. Here the long-term elements play a negligible role, while the transitional

factors are leading. This is in conformity with the rather stormy developments in East-

Germany when compared to those in the western part of the country.

Table 4a. Variance decomposition of income satisfaction inequality according to maximizing order. In percentages. Variable West Workers East Workers West

Non-Workers East

Non-Workers Ln2(age) 2.59% 1.30% 2.62% 1.62% Ln(age) 3.40% 11.54% 14.97% 22.03% Ln(net family inc.) 0.87% 56.69% 0.86% 26.43% Male 0.17% 1.05% 3.48% 1.14% Living together? 2.07% 1.75% 3.06% 2.42% Ln(years Education) 1.17% 0.01% 0.76% 8.68% ln(net fam.inc.) *ln(child.+1)

0.15% 0.00% 0.56% 0.00%

Ln(adults) 1.26% 11.44% 0.12% 22.17% Ln(Child.+1) 0.17% 8.16% 0.08% 13.95% Earner 0.05% 0.59% XXX XXX Mean(Ln(net fam. Inc.)) 65.63% 6.06% 35.13% 0.62% Mean(Ln(Child.+1)) 10.25% 0.01% 21.73% 0.26% Mean(ln(adults)) 12.21% 1.41% 16.62% 0.69%

In Table 4.b we present the corresponding decompositions, where we fix the order for

each sample as the order of variables in Table 2.

18

Table 4b. Variance decomposition of income satisfaction inequality according to order in Table 2. In percentages. West

Workers East Workers West

Non-Workers East

Non-Workers Ln(age) 0.12% 5.68% 9.18% 2.72% Ln2(age) 1.34% 1.77% 23.41% 50.06% Ln(net family inc.) 56.60% 61.55% 42.99% 29.68% Ln(years Education) 6.48% 1.32% 1.43% 0.68% Ln(adults) 10.32% 14.20% 5.75% 11.98% Ln(Child.+1) 4.77% 4.83% 2.51% 1.94% ln(net fam.inc.) *ln(child.+1)

0.18% 0.02% 0.09% 0.00%

Male 0.28% 1.12% 3.88% 1.00% Living together? 3.41% 2.06% 3.63% 0.38% Earner 0.21% 0.94% 0.00% Mean(Ln(net fam. Inc.)) 13.12% 5.14% 5.05% 0.62% Mean(Ln(Child.+1)) 0.48% 1.37% 0.23% 0.68% Mean(ln(adults)) 2.68% 0.01% 1.85% 0.25%

When we compare Tables 4a. and 4b., we find that the order in which the different

variables are brought into play is very important. Actually, the main burden of the

inequality shifts in Table 4b. to the transitory factors, although the explanation of the

long-term factors being about 16 % is still larger than the 12 % given to the transitory

factors in Table 4a. For non-workers we find that age becomes a very important factor

determining inequality.

Finally, we may take a look at income satisfaction inequality in the whole of

Germany (G). We use the well-known variance decomposition formula, where total

variance is split up into the sum of the two within-group-variances and the between-

groups variance. The two groups are the West-and East-German population in this

case. More precisely we have the identity

( ) ( ) ( ) ( )sat w sat E sat satI G p I W p I E I BetweenEandW= + + (5)

where the p’s stand for the relative population shares. The last term is calculated by

taking the variance of the mean of Western log-income satisfaction and the mean of

Eastern log-income satisfaction with respect to the overall mean log-income

satisfaction. In a similar way we may go on and decompose Isat(W) and Isat(E) with

19

respect to workers and non-workers. That decomposition is tabulated in Table 5. The

results are comparable to those presented in Table 3.

Table 5. Between-group decompositions for Income Satisfaction Inequality (Isat) Population

Shares Group Group Variance of structural

log-income satisfaction

PW = 0.803 West 0.264 PWW = 0.549 West Workers (WW) 0.186 PWNW =0.451 West Non-Workers (WNW) 0.357 Between WW and WNW 0.001 PE =0.197 East 0.265 PEW =0.528 East Workers (EW) 0.141 PENW =0.472 East Non-Workers (ENW) 0.328 Between EW and ENW 0.036 Between E and W 0.010 Germany 0.274

Table 5 shows that the income satisfaction inequality in Germany as a whole is 0.274,

and that there is virtually no difference between the two inequalities in East and West.

The income satisfaction distribution of Non-workers is more unequal than for workers

in both parts of the country.

The same exercise may be done for the objective income inequality. The results

are presented in Table 6. Again, the reader can compare these results with the ones

presented at Table 5. Table 6 illustrates that the objective income inequality in

Germany as a whole is 0.259, which is somewhat smaller than the income satisfaction

inequality. The Westerners suffer from a larger inequality than in the East and the

same holds for the corresponding subgroups of workers and non-workers..

Table 6. Between group decompositions for Income Inequality Population

Shares Group Group Variance of objective

Log-incomes PW = 0.803 West 0.261 PWW = 0.549 West Workers 0.218 PWNW =0.451 West Non-Workers 0.284 Between WW and WNW 0.0132 PE =0.197 East 0.219 PEW =0.528 East Workers 0.173 PENW =0.472 East Non-Workers 0.218 Between EW and ENW 0.0248 Between E and W 0.0063 Germany 0.259

20

Recently Gary Fields (2002) suggested another decomposition, which may be

succinctly described as follows. If there holds 1 1 ... k kZ Z Zβ β ε= + + + , there also

holds

1 1var( ) cov( , ) ... cov( , ) var( , )k kZ Z Z Z Z Zβ β ε= + + +

Division by var( )Z yields

1 ... 1kp p pε+ + + =

This is another way to define inequality or variance shares. The advantage of this

decomposition is that the shares do not depend on the order of introduction of the

variables. They are uniquely defined, thereby avoiding the element of arbitrariness,

which is inherent to the stepwise procedure, followed above. However, the price to be

paid is that the shares p may be negative or larger than one, which makes

interpretation cumbersome. It is therefore, that we did not use the Fields-

decomposition.

5. Conclusions

In this paper we extended the objective income concept by defining the subjective

income satisfaction concept. Similarly we extend the objective income inequality

concept by defining an income satisfaction inequality concept. The Isat measure differs

from objective measures of inequality as individual subjective satisfaction with

income is used instead of objective income. In other words, the paper presents

estimates for feelings of income inequality. The measure Isat includes objective income

inequality as a special case, namely, when subjective income satisfaction and income

are identical.

We find that only a relatively small part of Isat can be attributed to observed

factors. This does not necessarily imply that there would be no other observable

causes of inequality. It may be that the specification presented in Table 2 omitted

relevant observable variables. Nevertheless, this is not very likely, given the large

range of variables available in the GSOEP and the extensive research we did trying

21

different possible specifications. Even if the variance due to observable factors is

rather small, it is interesting to look at it, given that the objective variables are the

only ones, which policy makers can take into account. The role of income in

explaining income satisfaction inequality is not insignificant but it is not the only

causing factor. The number of people in the household and the age distribution are

important as well. Thus, even if objective income inequality remains certainly an

important statistic to monitor the societal distribution process, this exercise shows that

psychological feelings of inequality are relevant as well. Evidently, this research

should be repeated for other populations, before we may generalize the findings of

this paper.

This paper contributes to the literature of inequality by presenting an income

satisfaction concept, which can be compared to objective measures of inequality.

Income satisfaction inequality differs from the established measures of inequality by

using individual perceptions as a basis to make incomes comparable. The traditional

measures of inequality introduce subjectivism via intuition by, for example, imposing

family equivalence scales (such as the Oxford/OECD scale) or by introspection in

choosing a concrete welfare function specification with a numerically determined

risk/inequality aversion parameter (Atkinson, 1970). The introduction of income

satisfaction does not imply that objective measurement should be replaced by

subjective concepts throughout, but only that both measures have a different role to

play. The subjective concept is in our opinion a valuable addition to the family of

inequality measures.

22

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